Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

3, -13, and 5 + 4i

Answer 1

A) f(x) = x^4 - 8x^3 - 12x^2 + 400x - 1599
B) f(x) = x^4 - 200x^2 + 800x - 1599
C) f(x) = x^4 - 98x^2 + 800x - 1599
D) f(x) = x^4 - 8x^3 + 12x^2 - 400x + 1599

Answer 2

If a polynomial with real coefficients has complex roots, they *always* come in complex conjugate pairs. In other words, if one of the roots is a+ b*i (where i = sqr(-1) ), there is also the root a - b*i

in your case, you have a 4th root (the complex conjugate of 5 + 4i ) = 5 - 4i

your polynomial in factored form is
( x - (5+4i)) (x-(5-4i)) (x-3) (x - -13)
=
( x - (5+4i)) (x-(5-4i)) (x-3) (x +13)

Answer 3

How did you do that last step @phi ?

Answer 4

the last step is to find a polynomial with zero of \(5+4i\) which you can do by working backwards
\[x=5+4i\\
x-5=4i\\
(x-5)^2=(4i)^2=-16\\
x^2-10x+25=-16\\
x^2-10x+41=0\]

Answer 5

or you can know that if \(a+bi\) is a zero of a quadratic, the quadratic is
\[x^2-2ax+(a^2+b^2)\]

Answer 6

Oh, okay.. Sooo.. I'm real slow right now. I don't want the answer, but I still can't quite understand this...

Answer 7

Using Sat's answer, you have
\[ ( x - (5+4i)) (x-(5-4i)) (x-3) (x +13) \\
= (x^2-10x+41) (x-3) (x +13) \]

To finish, see
https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/multiplying_polynomials/v/multiplying-polynomials
and the next 2 videos. See if you can do this problem.

Answer 8

This is going to sound real, real dumb.. How do you multiply three's? I forgot the term.. I am super slow today. Example: \[(x^2-10x+41) (x^2+10x-39)\]

Answer 9

@phi

Answer 10

it is sort of like multiplying 123 * 456
by hand, if you learned how to do that